On Huppert's $\rho$-$\sigma$ conjecture for primes in the orders of elements

Abstract

Let $G$ be a finite group, let $\rho(G)$ denote the number of distinct prime divisors of $|G|$, and let $\sigma(G)$ be the maximal number of distinct prime divisors of the order of an element of $G$. Let $\rho(n)=\max\{\rho(G) \mid G \text{ solvable}, \sigma(G)=n\}$. We prove that $\rho(n)\le 4n-1+\pi(2n)$ for all $n$, and, with the help of a computer, that $\rho(n)\le 9n/2$ for all $n\le 16$. In particular, $\rho(n)\le\lfloor 9n/2\rfloor+2$ and $\rho(n)\le (101/22)n$ for all $n$, and $\rho(n)\le 9n/2$ for all $n\ge 58$. This improves the bound $\rho(n)\le (17/3)n$ due to Yang, and it shows that the bound $\rho(n)\le 5n$ stated by Bellotti, Keller and Trudgian holds, although its original proof contains a gap. Moreover, we show that $\rho(n)\ge 3n$ for all $n\ge 4$, which proves one half of a conjecture of the second author stating that $\rho(n)=3n$ for $n\ge 4$. We also derive a consequence for character codegrees and, conditionally, for arbitrary finite groups. The proofs of the main results do not rely on the classification of finite simple groups.